What is so wrong with thinking of real numbers as infinite decimals? | Hacker News Reader

What is so wrong with thinking of real numbers as infinite decimals?

dpmms.cam.ac.uk

18 pointstokenadult15 comments

> However, it has the advantage, for beginners, of being very close to the picture of real numbers they already have.

Haha, no. And here's why.

> Because of irritating difficulties such as the need to carry digits and to identify 0.999999.... with 1

Not only does this not fit in with people's existing picture of real numbers, people become emotionally attached to the opposite with a fervor unknown in any other part of mathematics! (Second place would be Cantor's theorem for real numbers, but you have to know a little more to be passionately opposed to that.)

I think even for beginners who are willing to believe that 0.99999... = 1, they are going to have trouble agreeing that 0.99999... < 1 is a false statement.

"Beginners" here mostly means "first-year undergraduates reading mathematics at the University of Cambridge". I can assure you that those people very seldom have any trouble agreeing that "0.999... < 1" is false.

[EDITED to insert the word "mostly"; I expect WTG wasn't thinking only of his and his colleagues' first-year students.]

On a somewhat related note: as computer scientists, we're usually dealing with discrete data, so correctness comes naturally to us with discrete problems.

The problem with floating point numbers is that most mathematical operations on them are approximations. People expect the same kind of correctness from floating point arithmetic that we've come to expect from integer arithmetic, when you really should be thinking in error bars.

I mean, just ask the average programmer to do currency math and I assure you they'll go straight for the `double`. Most of the time, not a catastrophic error, except for when it is.

For the curious, this is a long but very thorough overview of floating point arithmetic and it's quirks: http://docs.sun.com/source/806-3568/ncg_goldberg.html

That's all very well (and Goldberg's paper is excellent); but it has nothing whatever to do with the linked page, which is by Fields medallist Tim Gowers, arguing that in pure mathematics "infinite decimals" is as good a way to construct the real numbers as "Cauchy sequences" or "Dedekind cuts".

For what little it's worth, I agree with Gowers, and have done since before I read that page, though I'd prefer to use binary rather than decimal.

Thanks for pointing that out, I stand corrected.

In my defense, I wasn't trying to criticize the theoretical content of the article, which, in all honesty, is beyond my mathematical knowledge to criticize. The real number to "infinite" decimal conversion issue just reminded me of the age-old floating point number arithmetic problem, so I let out a little rant.

Not just computer scientists. I think normal people are used to dealing with discrete data. Ask a person on the street to name some numbers and they'll almost certainly give you integers.

Also, I'm not a programmer, I'm a mathematician. Is the "right" way to do currency math do to integer arithmetic on prices expressed in cents (or, more generally, the smallest subdivision of the currency)?

An integer expressing dollars, cents, or hundredths of a cent depending on the level of precision you need is the usual best practice.

Using floating point math for money is just asking for trouble.

Also see:

http://speleotrove.com/decimal

The bottom line is. There are numerous, equivalent ways of defining the real numbers and each have the advantages and disadvantages in understanding. Any good introduction to analysis class will do a definition several times and prove equivalence.

Infinite decimals in particular bring up the equality problem. If you're willing to think of the real numbers as ONLY infinite decimals, you avoid this confusion. But it's silly to always write .999... instead of 1.

Linked article reminded me of a passage in "Topoi - The Categorial Analysis of Logic" by Robert Goldblatt:

"It would be somewhat misleading to infer [...] that foundational systems act primarily as a basis out of which mathematics is actually created. The artificiality of that view is evident when one reflects that the essential content of mathematics is already there before the basis is made explicitly, and does not depend on it for its existence. We may for example think of a real number as an infinite decimal expression, or a point on the number line. Alternatively it could be introduced as an element of a complete ordered field, an equivalence class of Cauchy sequences, or a Dedekind cut. None of these could be said to be the correct explanation of what a real number is. Each is an embodiment of an intuitive notion and we evaluate it, not in terms of its correctness, but rather in terms of its effectiveness in explicating the nature of the real number system."

Tim Gowers makes a good case for the usefulness of the "infinite decimal expression" view of real numbers.

Gowers has a few dozen short pieces of this sort, listed at http://www.dpmms.cam.ac.uk/~wtg10/mathsindex.html. Another one that might be of interest is "why study finite-dimensional vector spaces in the abstract if they are all isomorphic to R^n?" (http://www.dpmms.cam.ac.uk/~wtg10/vspaces.html) The answer, roughly, is that the notation is less ugly if you don't have to keep track of coordinates, and that the abstract theory carries over better to the infinite-dimensional case.

> "why study finite-dimensional vector spaces in the abstract if they are all isomorphic to R^n?"

Uh, C^2?

That's already been posted to HN in the last few days.

Cantor went mad thinking about infinity - to me, a clear sign of how worthwhile the OCD path is.

To quote my favorite close-is-good-enough source, Wikipedia, "Poincaré referred to Cantor's ideas as a "grave disease" infecting the discipline of mathematics".

There's nothing wrong with thinking of the real numbers as infinite decimals. But you probably already do. Learning to think of them as the limits of cauchy sequences is good exercise for learning to think like a mathematician.

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